scholarly journals From Binary Groups to Terminal Rings

2021 ◽  
Author(s):  
Stuart D. Scott
Keyword(s):  
Special Interest ◽  
Associative Ring ◽  
Polynomial Maps ◽  
Finite Case ◽  
Restricted Form ◽  
Odd Order ◽  
Binary Group ◽  
Associative Rings

Binary groups are a meaningful step up from non-associative rings and nearrings. It makes sense to study them in terms of their nearrings of zero-fixing polynomial maps. As this involves algebras of a more specialized nature these are looked into in sections three and four. One of the main theorems of this paper occurs in section five where it is shown that a binary group V is a P0(V) ring module if, and only if, it is a rather restricted form of non-associative ring. Properties of these non-associative rings (called terminal rings) are investigated in sections six and seven. The finite case is of special interest since here terminal rings of odd order really are quite restricted. Sections eight to thirteen are taken up with the study of terminal rings of order pn (p an odd prime and n ≥ 1 an integer ≤ 7).

Zariski-like topologies for lattices with applications to modules over associative rings

2019 ◽  
Vol 18 (07) ◽  
pp. 1950131
Author(s):  
Jawad Abuhlail ◽  
Hamza Hroub
Keyword(s):  
Complete Lattice ◽  
Associative Ring ◽  
Sufficient Conditions ◽  
Proper Class ◽  
Topological Properties ◽  
Left Module ◽  
Separation Axioms ◽  
Prime Submodules ◽  
Algebraic Properties ◽  
Associative Rings

We study Zariski-like topologies on a proper class [Formula: see text] of a complete lattice [Formula: see text]. We consider [Formula: see text] with the so-called classical Zariski topology [Formula: see text] and study its topological properties (e.g. the separation axioms, the connectedness, the compactness) and provide sufficient conditions for it to be spectral. We say that [Formula: see text] is [Formula: see text]-top if [Formula: see text] is a topology. We study the interplay between the algebraic properties of an [Formula: see text]-top complete lattice [Formula: see text] and the topological properties of [Formula: see text] Our results are applied to several spectra which are proper classes of [Formula: see text] where [Formula: see text] is a nonzero left module over an arbitrary associative ring [Formula: see text] (e.g. the spectra of prime, coprime, fully prime submodules) of [Formula: see text] as well as to several spectra of the dual complete lattice [Formula: see text] (e.g. the spectra of first, second and fully coprime submodules of [Formula: see text]).

scholarly journals Weakly prime left ideals in general rings

1985 ◽  
Vol 32 (3) ◽  
pp. 357-360
Author(s):  
Halina France-Jackson
Keyword(s):  
Left Ideal ◽  
Associative Ring ◽  
Almost All ◽  
Associative Rings

A.P.J. van der Walt introduced the concept of a weakly prime left ideal of an associative ring with unity. It is the purpose of the present paper to extend to general, that is not necessarily with unity associative rings, this concept as well as almost all results of van der Walt for rings with unity.

Commutators in associative rings

1953 ◽  
Vol 49 (4) ◽  
pp. 590-594 ◽  
Author(s):  
M. P. Drazin ◽  
K. W. Gruenberg
Keyword(s):  
Associative Ring ◽  
Lie Ring ◽  
Addition And Subtraction ◽  
Associative Rings

Let R be an arbitrary associative ring, and X a set of generators of R. The elements of X generate a Lie ring, [X], say, with respect to the addition and subtraction in R, and the multiplication [a, b] = ab − ba. In this note we shall be concerned with the following question: if [X] is given to be nilpotent as a Lie ring, what does this imply about R?

scholarly journals Ideals in simple rings

1978 ◽  
Vol 25 (3) ◽  
pp. 322-327
Author(s):  
W. Harold Davenport
Keyword(s):  
Lie Algebra ◽  
Associative Ring ◽  
Alternative Ring ◽  
Cayley Algebra ◽  
Characteristic 3 ◽  
Associative Rings ◽  
Algebra Over A Field

AbstractIn this article, we define the concept of a Malcev ideal in an alternative ring in a manner analogous to Lie ideals in associative rings. By using a result of Kleinfield's we show that a nonassociative alternative ring of characteristic not 2 or 3 is a ring sum of Malcev ideals Z and [R, R] where Z is the center of R and [R, R] is a simple non-Lie Malcev ideal of R. If R is a Cayley algebra over a field F of characteristic 3 then [R, R] is a simple 7 dimensional Lie algebra. A similar result is obtained if R is a simple associative ring.

FINITE HIGHER COMMUTATORS IN ASSOCIATIVE RINGS

2013 ◽  
Vol 89 (3) ◽  
pp. 503-509
Author(s):  
CHARLES LANSKI
Keyword(s):  
Nilpotent Element ◽  
Associative Ring ◽  
Minimal Cardinality ◽  
Associative Rings ◽  
The Ideal

AbstractIf $T$ is any finite higher commutator in an associative ring $R$, for example, $T= [[R, R] , [R, R] ] $, and if $T$ has minimal cardinality so that the ideal generated by $T$ is infinite, then $T$ is in the centre of $R$ and ${T}^{2} = 0$. Also, if $T$ is any finite, higher commutator containing no nonzero nilpotent element then $T$ generates a finite ideal.

scholarly journals Direct Product of Finite Intuitionistic Anti Fuzzy Normal Subrings over Non-associative Rings

2019 ◽  
Vol 12 (2) ◽  
pp. 622-648 ◽  
Author(s):  
Nasreen Kausar
Keyword(s):  
Direct Product ◽  
Associative Ring ◽  
Intuitionistic Fuzzy ◽  
Associative Rings

Shal et. al cite:SKR, have introduced the concept of intuitionistic fuzzy normal subrings over a non-associative ring. In this paper, we investigate the concept of intuitionistic anti fuzzy normal subrings over non-associative rings and give some properties of such subrings

The compressed zero-divisor graphs of order 4

2021 ◽  
pp. 2250179
Author(s):  
A. S. Monastyreva
Keyword(s):  
Associative Ring ◽  
Zero Divisor ◽  
Finite Rings ◽  
Associative Rings ◽  
Siberian Math

In [E. V. Zhuravlev and A. S. Monastyreva, Compressed zero-divisor graphs of finite associative rings, Siberian Math. J. 61(1) (2020) 76–84.], we found the graphs containing at most three vertices that can be realized as the compressed zero-divisor graphs of some finite associative ring. This paper deals with associative finite rings whose compressed zero-divisor graphs have four vertices. Namely, we find all graphs containing four vertices that can be realized as the compressed zero-divisor graphs of some finite associative ring.

scholarly journals A Note on Additive Groups of Some Specific Associative Rings

2016 ◽  
Vol 30 (1) ◽  
pp. 219-229
Author(s):  
Mateusz Woronowicz
Keyword(s):  
Associative Ring ◽  
Abelian Groups ◽  
Free Groups ◽  
Additive Group ◽  
Direct Sums ◽  
Rank One ◽  
Elementary Methods ◽  
Associative Rings ◽  
Torsion Free

AbstractAlmost complete description of abelian groups (A, +, 0) such that every associative ring R with the additive group A satisfies the condition: every subgroup of A is an ideal of R, is given. Some new results for SR-groups in the case of associative rings are also achieved. The characterization of abelian torsion-free groups of rank one and their direct sums which are not nil-groups is complemented using only elementary methods.

scholarly journals ABSENCE OF TORSION FOR NK1(R) OVER ASSOCIATIVE RINGS

2011 ◽  
Vol 10 (04) ◽  
pp. 793-799 ◽  
Author(s):  
RABEYA BASU
Keyword(s):  
Commutative Ring ◽  
Associative Ring ◽  
Module Structure ◽  
Lecture Notes ◽  
Associative Rings

When R is a commutative ring with identity, and if k ∈ ℕ, with kR = R, then it was shown in [C. Weibel, Mayer–Vietoris Sequence and Module Structure on NK0, Lecture Notes in Mathematics, Vol. 854 (Springer, 1981), pp. 466–498] that SK 1(R[X]) has no k-torsion. We prove this result for any associative ring R with identity in which kR = R.

scholarly journals A commutativity theorem for power-associative rings

1970 ◽  
Vol 3 (1) ◽  
pp. 75-79 ◽  
Author(s):  
D. L. Outcalt ◽  
Adil Yaqub
Keyword(s):  
Finite Field ◽  
Associative Ring ◽  
Commutativity Theorem ◽  
Associative Rings

Let R be a power-associative ring with identity and let I be an ideal of R such that R/I is a finite field and x ≡ y (mod I) implies x2 = y2 or both x and y commute with all elements of I. It is proven that R must then be commutative. Examples are given to show that R need not be commutative if various parts of the hypothesis are dropped or if “x2 = y2” is replaced by “xk = yk” for any integer k > 2.

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